Elements Of Modern Algebra

Label each of the following statements as either true or false. Every mapping on a nonempty set A is a relation.

1TFE

Whether the statement, “Every mapping on the nonempty set A is a relation” is true or false.

Whether the statement, &#8220;Every mapping on the nonempty set <m:math><m:mi>A</m:mi></m:math> is a relation&#8221; is true or false.

Solution: The statement, “Every mapping on the nonempty set A is a relation” is true .

Solution:
The statement, &#8220;Every mapping on the nonempty set <m:math><m:mi>A</m:mi></m:math> is a relation&#8221; is true.

Consider the statement, “Every mapping on the nonempty set A is a relation”. Let A and B are non-empty sets. A subset f of A × B is a mapping from A to B if and only if for every a ∈ A there exists unique(one and only one) image b ∈ B such that ( a , b ) ∈ f A relation on a nonempty set A is a nonempty set R of ordered pairs ( x , y ) of elements of x and y of A . From the above definitions, Every mapping is a relation. Hence the statement, “Every mapping on the nonempty set A is a relation” is true.

Consider the statement, &#8220;Every mapping on the nonempty set <m:math><m:mi>A</m:mi></m:math> is a relation&#8221;.
Let <m:math><m:mi>A</m:mi></m:math> and <m:math><m:mi>B</m:mi></m:math> are non-empty sets. A subset <m:math><m:mi>f</m:mi></m:math> of <m:math><m:mrow><m:mi>A</m:mi><m:mo>&#x00D7;</m:mo><m:mi>B</m:mi></m:mrow></m:math> is a mapping from <m:math><m:mi>A</m:mi></m:math> to <m:math><m:mi>B</m:mi></m:math> if and only if for every <m:math><m:mrow><m:mi>a</m:mi><m:mo>&#x2208;</m:mo><m:mi>A</m:mi></m:mrow></m:math> there exists unique(one and only one) image <m:math><m:mrow><m:mi>b</m:mi><m:mo>&#x2208;</m:mo><m:mi>B</m:mi></m:mrow></m:math> such that <m:math><m:mrow><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>a</m:mi><m:mo>,</m:mo><m:mtext>&#x2009;</m:mtext><m:mtext>&#x2009;</m:mtext><m:mi>b</m:mi></m:mrow><m:mo>)</m:mo></m:mrow><m:mo>&#x2208;</m:mo><m:mi>f</m:mi></m:mrow></m:math>
A relation on a nonempty set <m:math><m:mi>A</m:mi></m:math> is a nonempty set <m:math><m:mi>R</m:mi></m:math> of ordered pairs <m:math><m:mrow><m:mrow><m:mo>(</m:mo><m:mrow><m:mi>x</m:mi><m:mo>,</m:mo><m:mtext>&#x2009;</m:mtext><m:mtext>&#x2009;</m:mtext><m:mi>y</m:mi></m:mrow><m:mo>)</m:mo></m:mrow></m:mrow></m:math> of elements of <m:math><m:mi>x</m:mi></m:math> and <m:math><m:mi>y</m:mi></m:math> of <m:math><m:mi>A</m:mi></m:math>.
From the above definitions,
Every mapping is a relation.
Hence the statement, &#8220;Every mapping on the nonempty set <m:math><m:mi>A</m:mi></m:math> is a relation&#8221; is true.

Label each of the following statements as either true or false.

Every mapping on a nonempty set A is a relation.

Answer

Explanation of Solution